Massive MIMO (multiple-input and multiple-output) is an emerging technology where the number of mobile terminals (UE) is much less than the number of base station antennas. In a rich scattering environment, the full advantages of the massive MIMO system can be exploited using simple beamforming strategies such as maximum ratio transmission (MRT) or zero forcing (ZF). To achieve these benefits of massive MIMO, accurate channel state information (CSI) must be available perfectly. However, in practice, the channel between the transmitter and receiver is estimated from orthogonal pilot sequences which are limited by the coherence time of the channel. Most importantly, in a multicell setup, the reuse of pilot sequences of several co-channel cells will create pilot contamination. When there is pilot contamination, the performance of massive MIMO degrades quite drastically.
If a base station (BS) is equipped with a large antenna number, there will be special aspect called channel hardening, which means the small scale fading is smoothed out by the asymptotically infinite number of antennas. Channel hardening refers to the phenomenon where the off-diagonal terms of the HHH matrix become increasingly weaker compared to the diagonal terms as the size of the channel gain matrix H increases.
For example, if one assumes each of all the L cells has K UEs (user equipment) for different cells their k-th UE use the same pilot slot, then the capacity with antenna number M approaching infinity can be expressed as
  I  =      log    ⁢                  ⁢                  det        ⁡                  (                                    I              M                        +                                          ρ                M                            ⁢                              HH                H                                              )                    ⁢              ⟶                  M          →          ∞                    ⁢      N        ⁢                  ⁢          log      ⁡              (                  1          +          ρ                )            because
                                          HH            H                    M                ⁢                  ⟶                      M            →            ∞                          ⁢        I                            (        1        )            
where the scalar ρ is the transmit power, M is the number of the BS antenna, H is the channel gain matrix, (⋅)H denotes the Hermitian matrix. When the number of transmit antennas M goes to infinity, the row vectors of H are asymptotically orthogonal, and hence one has
                    HH        H            M        ⁢          ⟶              M        →        ∞              ⁢    I    .Small scale fading parts disappear. Only large-scale fading (path-loss and shadowing) remains. They form the major part of pilot contamination.
To alleviate the effect of pilot contamination, one can only pay attention to this large scale fading (path-loss and shadowing). Fortunately, there is a partial connectivity feature in this large scale fading in practical systems. Not all the inter-cell interference links are strong enough to be attached equal importance. Only part of the links create strong interference, while others are negligible. Referring to FIG. 1, for a BS in a cell 1, the UEs on the cell edge will create strong inter-cell interference, but some UEs in cell 2 far from the BS only introduce negligible interference.
For example, if one has 4 BSs, 4 UEs (1 UE per BS) and 4 time slots for pilot, then the Tx (transmitter) pilot and Rx (receiver) projection design can be simple as shown in FIG. 2. One only needs to assign each UE one orthogonal slot. Referring to FIG. 2, Φ is a BS projection matrix in which the row number equals to the BS number 4, and the column number equals to the time slot number, each row represents the 4 time slots projection coefficient for each BS. Ψ is a UE Pilot matrix in which the row number equals to the time slot number, and the column number equals to the UE number, each column represents the 4 time slots pilot signal for each UE. So in this example, the BS projection matrix is Φ4×4, and the UE Pilot matrix is Ψ4×4.
However, in practice, the number of UE is always larger than the number of pilot slot. For example, if one has 6 BSs, 6 UEs (1 UE per BS) and 4 time slots for pilot, then the previous orthogonal pilot is unavailable. So in this example, the BS projection matrix is Φ6×4, and the UE Pilot matrix is Ψ4×6. The pilots have to be overlapping in time resources as shown in FIG. 6.
In a TDD system, all the users in all the cells first synchronously send uplink data signals. Next, the users send pilot sequences. BSs use these pilot sequences to estimate CSI to the users located in their cells. Then, BSs use the estimated CSI to detect the uplink data and to generate beamforming vectors for downlink data transmission. However, due to the limited channel coherence time, the pilot sequences employed by users in neighboring cells may no longer be orthogonal to those within the cell, leading to a pilot contamination problem (FIG. 3).
For TDD-based massive MIMO transmission systems, pilot sequences are transmitted from users in the uplink to estimate channels. Let Ψk,l=(ψk,l[1], . . . , ψk,l[τ])T be the pilot sequence of user k in cell l, where τ denotes the length of the pilot sequence. Though it is not necessary, it is convenient to assume that and one uses this assumption in what follows. Ideally, the pilot sequences employed by users within the same cell and in the neighboring cells should be orthogonal, that isΨk,lHΨj,l′=δ[k−j]δ[l−l′]  (2)where δ[.] is defined as
                              δ          ⁡                      [            n            ]                          =                  {                                                    1                                                              n                  =                  0                                                                                    0                                            otherwise                                                                        (        3        )            
In this case, a BS can obtain uncontaminated estimation of the channel vectors in the sense that they are not correlated to the channel vectors of other users.
However, the number of orthogonal pilot sequences with a given period and bandwidth is limited, which in turn limits the number of users that can be served. In order to handle more users, non-orthogonal pilot sequences are used in neighboring cells. Thus for some different k, j, l, and l′, one may haveΨk,lHΨj,l′≠0  (4)
As a result, the estimate of the channel vector to a user becomes correlated with the channel vectors of the users with non-orthogonal pilot sequences.
In summary, in a typical multi-cell massive MIMO system, users from neighboring cells may use non-orthogonal pilots. The reason for this is very simple—the number of orthogonal pilots is smaller than the number of users. The use of non-orthogonal pilots results in the pilot contamination problem. Pilot contamination causes directed inter-cell interference, which, unlike other sources of interferences, grows together with the number of BS antennas and significantly damages the system performance. Various channel estimation, precoding, and cooperation methods have been proposed to resolve this issue. However, more efficient methods with good performance, low complexity, and limited or zero cooperation between BSs are worth more intensive study.